American Institute of Mathematical Sciences

We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution \begin{document}$ (f_t)_{t\geq 0} $\end{document}, once the initial condition \begin{document}$ f_0 $\end{document} with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a \begin{document}$ (0,\infty)\times {\mathbb{R}}^3 $\end{document} random variable \begin{document}$ (M_t,V_t) $\end{document} such that \begin{document}$ \mathbb{E}[M_t {\bf 1}_{\{V_t \in \cdot\}}] = f_t $\end{document}. We also write down a series expansion of \begin{document}$ f_t $\end{document}. Although both the algorithm and the series expansion might be theoretically interesting in that they explicitly express \begin{document}$ f_t $\end{document} in terms of \begin{document}$ f_0 $\end{document}, we believe that the algorithm is not very efficient in practice and that the series expansion is rather intractable. This is a tedious extension to non-Maxwellian molecules of Wild's sum [18] and of its interpretation by McKean [10,11].

In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

We study the asymptotic behaviour of the following linear growth-fragmentation equation

\begin{document}$ \frac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{ \partial x} \big(x u(t,x)\big) + B(x) u(t,x) = 4 B(2x)u(t,2x), $\end{document}

and prove that under fairly general assumptions on the division rate \begin{document}$ B(x), $\end{document} its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. Despite the lack of hypocoercivity of the operator, the proof relies on a general relative entropy argument in a convenient weighted \begin{document}$ L^2 $\end{document} space, where well-posedness is obtained via semigroup analysis. We also propose a non-diffusive numerical scheme, able to capture the oscillations.

In this paper, we consider the Cucker-Smale flocking particles which are subject to the same velocity-dependent noise, which exhibits a phase change phenomenon occurs bringing the system from a "non flocking" to a "flocking" state as the strength of noises decreases. We rigorously show the stochastic mean-field limit from the many-particle Cucker-Smale system with multiplicative noises to the Vlasov-type stochastic partial differential equation as the number of particles goes to infinity. More precisely, we provide a quantitative error estimate between solutions to the stochastic particle system and measure-valued solutions to the expected limiting stochastic partial differential equation by using the Wasserstein distance. For the limiting equation, we construct global-in-time measure-valued solutions and study the stability and large-time behavior showing the convergence of velocities to their mean exponentially fast almost surely.

We consider various sets of Vlasov-Fokker-Planck equations modeling the dynamics of charged particles in a plasma under the effect of a strong magnetic field. For each of them in a regime where the strength of the magnetic field is effectively stronger than that of collisions we first formally derive asymptotically reduced models. In this regime, strong anisotropic phenomena occur; while equilibrium along magnetic field lines is asymptotically reached our asymptotic models capture a non trivial dynamics in the perpendicular directions. We do check that in any case the obtained asymptotic model defines a well-posed dynamical system and when self consistent electric fields are neglected we provide a rigorous mathematical justification of the formally derived systems. In this last step we provide a complete control on solutions by developing anisotropic hypocoercive estimates.

We investigate the time-asymptotic stability of planar rarefaction wave for the three-dimensional Boltzmann equation, based on the micro-macro decomposition introduced in [24,22] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of planar rarefaction wave for 3D Boltzmann equation, while the corresponding results for the shock and contact discontinuities are still completely open.